1.
Hexagon
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In geometry, a hexagon is a six sided polygon or 6-gon. The total of the angles of any hexagon is 720°. A regular hexagon has Schläfli symbol and can also be constructed as an equilateral triangle, t. A regular hexagon is defined as a hexagon that is both equilateral and equiangular and it is bicentric, meaning that it is both cyclic and tangential. The common length of the sides equals the radius of the circumscribed circle, all internal angles are 120 degrees. A regular hexagon has 6 rotational symmetries and 6 reflection symmetries, the longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. Like squares and equilateral triangles, regular hexagons fit together without any gaps to tile the plane, the cells of a beehive honeycomb are hexagonal for this reason and because the shape makes efficient use of space and building materials. The Voronoi diagram of a triangular lattice is the honeycomb tessellation of hexagons. It is not usually considered a triambus, although it is equilateral, the maximal diameter, D is twice the maximal radius or circumradius, R, which equals the side length, t. The minimal diameter or the diameter of the circle, d, is twice the minimal radius or inradius. If a regular hexagon has successive vertices A, B, C, D, E, F, the regular hexagon has Dih6 symmetry, order 12. There are 3 dihedral subgroups, Dih3, Dih2, and Dih1, and 4 cyclic subgroups, Z6, Z3, Z2 and these symmetries express 9 distinct symmetries of a regular hexagon. John Conway labels these by a letter and group order, r12 is full symmetry, and a1 is no symmetry. These two forms are duals of each other and have half the order of the regular hexagon. The i4 forms are regular hexagons flattened or stretched along one symmetry direction and it can be seen as an elongated rhombus, while d2 and p2 can be seen as horizontally and vertically elongated kites. G2 hexagons, with sides parallel are also called hexagonal parallelogons. Each subgroup symmetry allows one or more degrees of freedom for irregular forms, only the g6 subgroup has no degrees of freedom but can seen as directed edges. Hexagons of symmetry g2, i4, and r12, as parallelogons can tessellate the Euclidean plane by translation, other hexagon shapes can tile the plane with different orientations
2.
Pyramid
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A pyramid is a structure whose outer surfaces are triangular and converge to a single point at the top, making the shape roughly a pyramid in the geometric sense. The base of a pyramid can be trilateral, quadrilateral, or any polygon shape, as such, a pyramid has at least three outer triangular surfaces. The square pyramid, with base and four triangular outer surfaces, is a common version. A pyramids design, with the majority of the closer to the ground. This distribution of weight allowed early civilizations to create stable monumental structures and it has been demonstrated that the common shape of the pyramids of antiquity, from Egypt to Central America, represents the dry-stone construction that requires minimum human work. Pyramids have been built by civilizations in many parts of the world, khufus Pyramid is built mainly of limestone, and is considered an architectural masterpiece. It contains over 2,000,000 blocks ranging in weight from 2.5 tonnes to 15 tonnes and is built on a base with sides measuring about 230 m. Its four sides face the four cardinal points precisely and it has an angle of 52 degrees and it is still the tallest pyramid. The largest pyramid by volume is the Great Pyramid of Cholula, the Mesopotamians built the earliest pyramidal structures, called ziggurats. In ancient times, these were painted in gold/bronze. Since they were constructed of sun-dried mud-brick, little remains of them, ziggurats were built by the Sumerians, Babylonians, Elamites, Akkadians, and Assyrians for local religions. Each ziggurat was part of a complex which included other buildings. The precursors of the ziggurat were raised platforms that date from the Ubaid period during the fourth millennium BC, the earliest ziggurats began near the end of the Early Dynastic Period. The latest Mesopotamian ziggurats date from the 6th century BC, built in receding tiers upon a rectangular, oval, or square platform, the ziggurat was a pyramidal structure with a flat top. Sun-baked bricks made up the core of the ziggurat with facings of fired bricks on the outside, the facings were often glazed in different colors and may have had astrological significance. Kings sometimes had their names engraved on these glazed bricks, the number of tiers ranged from two to seven. It is assumed that they had shrines at the top, but there is no evidence for this. Access to the shrine would have been by a series of ramps on one side of the ziggurat or by a ramp from base to summit
3.
Tetrahedron
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In geometry, a tetrahedron, also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra, the tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a polygon base. In the case of a tetrahedron the base is a triangle, like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. For any tetrahedron there exists a sphere on which all four vertices lie, a regular tetrahedron is one in which all four faces are equilateral triangles. It is one of the five regular Platonic solids, which have known since antiquity. In a regular tetrahedron, not only are all its faces the same size and shape, regular tetrahedra alone do not tessellate, but if alternated with regular octahedra they form the alternated cubic honeycomb, which is a tessellation. The regular tetrahedron is self-dual, which means that its dual is another regular tetrahedron, the compound figure comprising two such dual tetrahedra form a stellated octahedron or stella octangula. This form has Coxeter diagram and Schläfli symbol h, the tetrahedron in this case has edge length 2√2. Inverting these coordinates generates the dual tetrahedron, and the together form the stellated octahedron. In other words, if C is the centroid of the base and this follows from the fact that the medians of a triangle intersect at its centroid, and this point divides each of them in two segments, one of which is twice as long as the other. The vertices of a cube can be grouped into two groups of four, each forming a regular tetrahedron, the symmetries of a regular tetrahedron correspond to half of those of a cube, those that map the tetrahedra to themselves, and not to each other. The tetrahedron is the only Platonic solid that is not mapped to itself by point inversion, the regular tetrahedron has 24 isometries, forming the symmetry group Td, isomorphic to the symmetric group, S4. The first corresponds to the A2 Coxeter plane, the two skew perpendicular opposite edges of a regular tetrahedron define a set of parallel planes. When one of these intersects the tetrahedron the resulting cross section is a rectangle. When the intersecting plane is one of the edges the rectangle is long. When halfway between the two edges the intersection is a square, the aspect ratio of the rectangle reverses as you pass this halfway point. For the midpoint square intersection the resulting boundary line traverses every face of the tetrahedron similarly, if the tetrahedron is bisected on this plane, both halves become wedges
4.
Polyhedron
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In geometry, a polyhedron is a solid in three dimensions with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron, a convex polyhedron is the convex hull of finitely many points, not all on the same plane. Cubes and pyramids are examples of convex polyhedra, a polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions. Convex polyhedra are well-defined, with several equivalent standard definitions, however, the formal mathematical definition of polyhedra that are not required to be convex has been problematic. Many definitions of polyhedron have been given within particular contexts, some more rigorous than others, some of these definitions exclude shapes that have often been counted as polyhedra or include shapes that are often not considered as valid polyhedra. As Branko Grünbaum observed, The Original Sin in the theory of polyhedra goes back to Euclid, the writers failed to define what are the polyhedra. Nevertheless, there is agreement that a polyhedron is a solid or surface that can be described by its vertices, edges, faces. Natural refinements of this definition require the solid to be bounded, to have a connected interior, and possibly also to have a connected boundary. However, the polyhedra defined in this way do not include the self-crossing star polyhedra, their faces may not form simple polygons, definitions based on the idea of a bounding surface rather than a solid are also common. If a planar part of such a surface is not itself a convex polygon, ORourke requires it to be subdivided into smaller convex polygons, cromwell gives a similar definition but without the restriction of three edges per vertex. Again, this type of definition does not encompass the self-crossing polyhedra, however, there exist topological polyhedra that cannot be realized as acoptic polyhedra. One modern approach is based on the theory of abstract polyhedra and these can be defined as partially ordered sets whose elements are the vertices, edges, and faces of a polyhedron. A vertex or edge element is less than an edge or face element when the vertex or edge is part of the edge or face, additionally, one may include a special bottom element of this partial order and a top element representing the whole polyhedron. However, these requirements are relaxed, to instead require only that the sections between elements two levels apart from line segments. Geometric polyhedra, defined in other ways, can be described abstractly in this way, a realization of an abstract polyhedron is generally taken to be a mapping from the vertices of the abstract polyhedron to geometric points, such that the points of each face are coplanar. A geometric polyhedron can then be defined as a realization of an abstract polyhedron, realizations that forgo the requirement of planarity, that impose additional requirements of symmetry, or that map the vertices to higher dimensional spaces have also been considered. Unlike the solid-based and surface-based definitions, this perfectly well for star polyhedra. However, without restrictions, this definition allows degenerate or unfaithful polyhedra
5.
Regular polyhedron
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A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive, in classical contexts, many different equivalent definitions are used, a common one is that faces are congruent regular polygons which are assembled in the same way around each vertex. A regular polyhedron is identified by its Schläfli symbol of the form, there are 5 finite convex regular polyhedra, known as the Platonic solids. These are the, tetrahedron, cube, octahedron, dodecahedron and icosahedron, there are also four regular star polyhedra, making nine regular polyhedra in all. All the dihedral angles of the polyhedron are equal All the vertex figures of the polyhedron are regular polygons, All the solid angles of the polyhedron are congruent. A regular polyhedron has all of three related spheres which share its centre, An insphere, tangent to all faces, an intersphere or midsphere, tangent to all edges. A circumsphere, tangent to all vertices, the regular polyhedra are the most symmetrical of all the polyhedra. They lie in just three symmetry groups, which are named after them, Tetrahedral Octahedral Icosahedral Any shapes with icosahedral or octahedral symmetry will also contain tetrahedral symmetry, the five Platonic solids have an Euler characteristic of 2. Some of the stars have a different value. The sum of the distances from any point in the interior of a polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra, in a dual pair of polyhedra, the vertices of one polyhedron correspond to the faces of the other, and vice versa. The regular polyhedra show this duality as follows, The tetrahedron is self-dual, the cube and octahedron are dual to each other. The icosahedron and dodecahedron are dual to each other, the small stellated dodecahedron and great dodecahedron are dual to each other. The great stellated dodecahedron and great icosahedron are dual to each other, the Schläfli symbol of the dual is just the original written backwards, for example the dual of is. See also Regular polytope, History of discovery, stones carved in shapes resembling clusters of spheres or knobs have been found in Scotland and may be as much as 4,000 years old. Some of these stones show not only the symmetries of the five Platonic solids, examples of these stones are on display in the John Evans room of the Ashmolean Museum at Oxford University. Why these objects were made, or how their creators gained the inspiration for them, is a mystery, the earliest known written records of the regular convex solids originated from Classical Greece. When these solids were all discovered and by whom is not known, euclids reference to Plato led to their common description as the Platonic solids
6.
Simply connected space
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If a space is not simply-connected, it is convenient to measure the extent to which it fails to be simply-connected, this is done by the fundamental group. Intuitively, the fundamental group measures how the holes behave on a space, if there are no holes, the group is trivial — equivalently. Informally, an object in our space is simply-connected if it consists of one piece. For example, neither a doughnut nor a cup is simply connected. In two dimensions, a circle is not simply-connected, but a disk and a line are, spaces that are connected but not simply connected are called non–simply connected or, in a somewhat old-fashioned term, multiply connected. To illustrate the notion of connectedness, suppose we are considering an object in three dimensions, for example, an object in the shape of a box, a doughnut. Think of the object as a strangely shaped aquarium full of water, with rigid sides. Now think of a diver who takes a piece of string and trails it through the water inside the aquarium, in whatever way he pleases. Now the loop begins to contract on itself, getting smaller and smaller, if the loop can always shrink all the way to a point, then the aquariums interior is simply connected. If sometimes the loop gets caught — for example, around the hole in the doughnut — then the object is not simply-connected. Notice that the only rules out handle-shaped holes. A sphere is connected, because any loop on the surface of a sphere can contract to a point. The stronger condition, that the object has no holes of any dimension, is called contractibility, intuitively, this means that p can be continuously deformed to get q while keeping the endpoints fixed. Hence the term simply connected, for any two points in X, there is one and essentially only one path connecting them. A third way to express the same, X is simply-connected if and only if X is path-connected and the fundamental group of X at each of its points is trivial, i. e. consists only of the identity element. Yet another formulation is used in complex analysis, an open subset X of C is simply-connected if. It might also be worth pointing out that a relaxation of the requirement that X be connected leads to an exploration of open subsets of the plane with connected extended complement. For example, a set has connected extended complement exactly when each of its connected components are simply-connected
7.
Projective plane
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In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines intersect in a single point, but there are some pairs of lines that do not intersect. A projective plane can be thought of as a plane equipped with additional points at infinity where parallel lines intersect. Thus any two lines in a projective plane intersect in one and only one point. Renaissance artists, in developing the techniques of drawing in perspective, the archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG, RP2. There are many projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces, such embeddability is a consequence of a property known as Desargues theorem, not shared by all projective planes. The last condition excludes the so-called degenerate cases, the term incidence is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression point P is incident with l is used instead of either P is on l or l passes through P. To turn the ordinary Euclidean plane into a projective plane proceed as follows and that point is considered incident with each line of the class. Different parallel classes get different points and these points are called points at infinity. Add a new line which is considered incident with all the points at infinity and this line is called the line at infinity. The extended structure is a plane and is called the Extended Euclidean Plane or the real projective plane. The process outlined above, used to obtain it, is called projective completion or projectivization and this plane can also be constructed by starting from R3 viewed as a vector space, see below. The points of the Moulton plane are the points of the Euclidean plane, to create the Moulton plane from the Euclidean plane some of the lines are redefined. That is, some of their point sets will be changed, the Moulton plane has parallel classes of lines and is an affine plane. It can be projectivized, as in the example, to obtain the projective Moulton plane
8.
Pentagonal prism
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In geometry, the pentagonal prism a prism with a pentagonal base. It is a type of heptahedron with 7 faces,15 edges and it can be seen as a truncated pentagonal hosohedron, represented by Schläfli symbol t. Alternately it can be seen as the Cartesian product of a pentagon and a line segment. The dual of a prism is a pentagonal bipyramid. The symmetry group of a pentagonal prism is D5h of order 20. The rotation group is D5 of order 10, the volume, as for all prisms, is the product of the area of the pentagonal base times the height or distance along any edge perpendicular to the base. It exists as cells of four nonprismatic uniform 4-polytopes in 4 dimensions, Weisstein, Pentagonal Prism Polyhedron Model -- works in your web browser
9.
Elongated triangular pyramid
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In geometry, the elongated triangular pyramid is one of the Johnson solids. As the name suggests, it can be constructed by elongating a tetrahedron by attaching a triangular prism to its base, like any elongated pyramid, the resulting solid is topologically self-dual. A Johnson solid is one of 92 strictly convex polyhedra that have regular faces but are not uniform and they were named by Norman Johnson, who first listed these polyhedra in 1966. The following formulae for volume and surface area can be used if all faces are regular, a 2 If the edges are not the same length, use the individual formulae for the tetrahedron and triangular prism separately, and add the results together. Topologically, the triangular pyramid is its own dual. Geometrically, the dual has seven irregular faces, one equilateral triangle, the elongated triangular pyramid can form a tessellation of space with square pyramids and/or octahedra. Eric W. Weisstein, Johnson solid at MathWorld
10.
Face (geometry)
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In solid geometry, a face is a flat surface that forms part of the boundary of a solid object, a three-dimensional solid bounded exclusively by flat faces is a polyhedron. In more technical treatments of the geometry of polyhedra and higher-dimensional polytopes, in elementary geometry, a face is a polygon on the boundary of a polyhedron. Other names for a polygonal face include side of a polyhedron, for example, any of the six squares that bound a cube is a face of the cube. Sometimes face is used to refer to the 2-dimensional features of a 4-polytope. With this meaning, the 4-dimensional tesseract has 24 square faces, some other polygons, which are not faces, are also important for polyhedra and tessellations. These include Petrie polygons, vertex figures and facets, any convex polyhedrons surface has Euler characteristic V − E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. This equation is known as Eulers polyhedron formula, thus the number of faces is 2 more than the excess of the number of edges over the number of vertices. For example, a cube has 12 edges and 8 vertices, in higher-dimensional geometry the faces of a polytope are features of all dimensions. A face of dimension k is called a k-face, for example, the polygonal faces of an ordinary polyhedron are 2-faces. In set theory, the set of faces of a polytope includes the polytope itself, for any n-polytope, −1 ≤ k ≤ n. For example, with meaning, the faces of a cube include the empty set, its vertices, edges and squares. Formally, a face of a polytope P is the intersection of P with any closed halfspace whose boundary is disjoint from the interior of P, from this definition it follows that the set of faces of a polytope includes the polytope itself and the empty set. In other areas of mathematics, such as the theories of abstract polytopes and star polytopes, abstract theory still requires that the set of faces include the polytope itself and the empty set. A cell is an element of a 4-dimensional polytope or 3-dimensional tessellation. Cells are facets for 4-polytopes and 3-honeycombs, examples, In higher-dimensional geometry, the facets of a n-polytope are the -faces of dimension one less than the polytope itself. A polytope is bounded by its facets, for example, The facets of a line segment are its 0-faces or vertices. The facets of a polygon are its 1-faces or edges, the facets of a polyhedron or plane tiling are its 2-faces. The facets of a 4D polytope or 3-honeycomb are its 3-faces, the facets of a 5D polytope or 4-honeycomb are its 4-faces
11.
Tetrahemihexahedron
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In geometry, the tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U4. It has six vertices,12 edges, and seven faces, four triangular and its vertex figure is a crossed quadrilateral. It is the only uniform polyhedron with an odd number of faces. Its Wythoff symbol is 3/23 |2, but that represents a double covering of the tetrahemihexahedron with eight triangles and six squares, paired, hemi faces are also oriented in the same direction as the regular polyhedrons faces. The three square faces of the tetrahemihexahedron are, like the three orientations of the cube, mutually perpendicular. The half-as-many characteristic also means that hemi faces must pass through the center of the polyhedron, visually, each square is divided into four right triangles, with two visible from each side. It is the three-dimensional demicross polytope and it has the same vertices and edges as the regular octahedron. It also shares 4 of the 8 triangular faces of the octahedron, the dual figure is the tetrahemihexacron. It is 2-covered by the cuboctahedron, which accordingly has the same vertex figure and twice the vertices, edges. It has the topology as the abstract polyhedron hemi-cuboctahedron. It may also be constructed as a crossed triangular cuploid, being a version of the -cupola by its -gonal base. The tetrahemihexacron is the dual of the tetrahemihexahedron, and is one of nine dual hemipolyhedra, since the hemipolyhedra have faces passing through the center, the dual figures have corresponding vertices at infinity, properly, on the real projective plane at infinity. In Magnus Wenningers Dual Models, they are represented with intersecting prisms, in practice the model prisms are cut off at a certain point that is convenient for the maker. Wenninger suggested these figures are members of a new class of stellation figures, however, he also suggested that strictly speaking they are not polyhedra because their construction does not conform to the usual definitions. Topologically it is considered to contain seven vertices, the three vertices considered at infinity correspond directionally to the three vertices of the hemi-octahedron, an abstract polyhedron. The other four vertices exist at alternate corners of a central cube, Eric W. Weisstein, Tetrahemihexahedron at MathWorld. Uniform polyhedra and duals Weisstein, Eric W. Tetrahemihexacron, paper model Great Stella, software used to create main image on this page